Question

If $A$ and $B$ are square matrices such that $B=-A^{-1}BA,$ then $(A+B{)}^2$ is equal to

• A. $0$
• B. $A^2+B^2$
• C. $A^2+2AB+B^2$
• D. $A+B$

Answer 1

The correct option is B $A^2+B^2$

$B=-A^{-1}BA$

Now pre multiplying by A on both sides, we will get

$AB=-BA$

$AB+BA=0$ ...(1)

Now

$(A+B{)}^2=(A+B\left)\right(A+B)$

$=A^2+B^2+BA+BA$

from (1) $AB+BA=0$

$(A+B{)}^2$

$=A^2{B}^2$